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In
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matric ...
, a multilinear map is a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orie ...
of several variables that is
linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
separately in each variable. More precisely, a multilinear map is a function :f\colon V_1 \times \cdots \times V_n \to W\text where V_1,\ldots,V_n and W are
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
s (or
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Mo ...
s over a commutative ring), with the following property: for each i, if all of the variables but v_i are held constant, then f(v_1, \ldots, v_i, \ldots, v_n) is a
linear function In mathematics, the term linear function refers to two distinct but related notions: * In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For di ...
of v_i. A multilinear map of one variable is a
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
, and of two variables is a
bilinear map In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example. Definition Vector spaces Let V, ...
. More generally, a multilinear map of ''k'' variables is called a ''k''-linear map. If the
codomain In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either ...
of a multilinear map is the
field of scalars In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
, it is called a
multilinear form In abstract algebra and multilinear algebra, a multilinear form on a vector space V over a field K is a map :f\colon V^k \to K that is separately ''K''-linear in each of its ''k'' arguments. More generally, one can define multilinear forms on ...
. Multilinear maps and multilinear forms are fundamental objects of study in
multilinear algebra Multilinear algebra is a subfield of mathematics that extends the methods of linear algebra. Just as linear algebra is built on the concept of a vector and develops the theory of vector spaces, multilinear algebra builds on the concepts of '' ...
. If all variables belong to the same space, one can consider
symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
, antisymmetric and
alternating Alternating may refer to: Mathematics * Alternating algebra, an algebra in which odd-grade elements square to zero * Alternating form, a function formula in algebra * Alternating group, the group of even permutations of a finite set * Alter ...
''k''-linear maps. The latter coincide if the underlying
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
(or
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
) has a characteristic different from two, else the former two coincide.


Examples

* Any
bilinear map In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example. Definition Vector spaces Let V, ...
is a multilinear map. For example, any
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
on a vector space is a multilinear map, as is the
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and i ...
of vectors in \mathbb^3. * The
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if ...
of a matrix is an
alternating Alternating may refer to: Mathematics * Alternating algebra, an algebra in which odd-grade elements square to zero * Alternating form, a function formula in algebra * Alternating group, the group of even permutations of a finite set * Alter ...
multilinear function of the columns (or rows) of a
square matrix In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Square matrices are ofte ...
. * If F\colon \mathbb^m \to \mathbb^n is a ''Ck'' function, then the k\!th derivative of F\! at each point p in its domain can be viewed as a
symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
k-linear function D^k\!F\colon \mathbb^m\times\cdots\times\mathbb^m \to \mathbb^n.


Coordinate representation

Let :f\colon V_1 \times \cdots \times V_n \to W\text be a multilinear map between finite-dimensional vector spaces, where V_i\! has dimension d_i\!, and W\! has dimension d\!. If we choose a
basis Basis may refer to: Finance and accounting *Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting o ...
\ for each V_i\! and a basis \ for W\! (using bold for vectors), then we can define a collection of scalars A_^k by :f(\textbf_,\ldots,\textbf_) = A_^1\,\textbf_1 + \cdots + A_^d\,\textbf_d. Then the scalars \ completely determine the multilinear function f\!. In particular, if :\textbf_i = \sum_^ v_ \textbf_\! for 1 \leq i \leq n\!, then :f(\textbf_1,\ldots,\textbf_n) = \sum_^ \cdots \sum_^ \sum_^ A_^k v_\cdots v_ \textbf_k.


Example

Let's take a trilinear function :g\colon R^2 \times R^2 \times R^2 \to R, where , and . A basis for each is \ = \ = \. Let :g(\textbf_,\textbf_,\textbf_) = f(\textbf_,\textbf_,\textbf_) = A_, where i,j,k \in \. In other words, the constant A_ is a function value at one of the eight possible triples of basis vectors (since there are two choices for each of the three V_i), namely: : \, \, \, \, \, \, \, \. Each vector \textbf_i \in V_i = R^2 can be expressed as a linear combination of the basis vectors :\textbf_i = \sum_^ v_ \textbf_ = v_ \times \textbf_1 + v_ \times \textbf_2 = v_ \times (1, 0) + v_ \times (0, 1). The function value at an arbitrary collection of three vectors \textbf_i \in R^2 can be expressed as :g(\textbf_1,\textbf_2, \textbf_3) = \sum_^ \sum_^ \sum_^ A_ v_ v_ v_. Or, in expanded form as : \begin g((a,b),(c,d)&, (e,f)) = ace \times g(\textbf_1, \textbf_1, \textbf_1) + acf \times g(\textbf_1, \textbf_1, \textbf_2) \\ &+ ade \times g(\textbf_1, \textbf_2, \textbf_1) + adf \times g(\textbf_1, \textbf_2, \textbf_2) + bce \times g(\textbf_2, \textbf_1, \textbf_1) + bcf \times g(\textbf_2, \textbf_1, \textbf_2) \\ &+ bde \times g(\textbf_2, \textbf_2, \textbf_1) + bdf \times g(\textbf_2, \textbf_2, \textbf_2). \end


Relation to tensor products

There is a natural one-to-one correspondence between multilinear maps :f\colon V_1 \times \cdots \times V_n \to W\text and linear maps :F\colon V_1 \otimes \cdots \otimes V_n \to W\text where V_1 \otimes \cdots \otimes V_n\! denotes the
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same Field (mathematics), field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an e ...
of V_1,\ldots,V_n. The relation between the functions f\! and F\! is given by the formula :f(v_1,\ldots,v_n)=F(v_1\otimes \cdots \otimes v_n).


Multilinear functions on ''n''×''n'' matrices

One can consider multilinear functions, on an matrix over a commutative ring with identity, as a function of the rows (or equivalently the columns) of the matrix. Let be such a matrix and , be the rows of . Then the multilinear function can be written as :D(A) = D(a_,\ldots,a_), satisfying :D(a_,\ldots,c a_ + a_',\ldots,a_) = c D(a_,\ldots,a_,\ldots,a_) + D(a_,\ldots,a_',\ldots,a_). If we let \hat_j represent the th row of the identity matrix, we can express each row as the sum :a_ = \sum_^n A(i,j)\hat_. Using the multilinearity of we rewrite as : D(A) = D\left(\sum_^n A(1,j)\hat_, a_2, \ldots, a_n\right) = \sum_^n A(1,j) D(\hat_,a_2,\ldots,a_n). Continuing this substitution for each we get, for , : D(A) = \sum_ \ldots \sum_ \ldots \sum_ A(1,k_)A(2,k_)\dots A(n,k_) D(\hat_,\dots,\hat_). Therefore, is uniquely determined by how operates on \hat_,\dots,\hat_.


Example

In the case of 2×2 matrices we get : D(A) = A_A_D(\hat_1,\hat_1) + A_A_D(\hat_1,\hat_2) + A_A_D(\hat_2,\hat_1) + A_A_D(\hat_2,\hat_2) \, Where \hat_1 = ,0/math> and \hat_2 = ,1/math>. If we restrict D to be an alternating function then D(\hat_1,\hat_1) = D(\hat_2,\hat_2) = 0 and D(\hat_2,\hat_1) = -D(\hat_1,\hat_2) = -D(I). Letting D(I) = 1 we get the determinant function on 2×2 matrices: : D(A) = A_A_ - A_A_{2,1} .


Properties

* A multilinear map has a value of zero whenever one of its arguments is zero.


See also

*
Algebraic form In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; t ...
*
Multilinear form In abstract algebra and multilinear algebra, a multilinear form on a vector space V over a field K is a map :f\colon V^k \to K that is separately ''K''-linear in each of its ''k'' arguments. More generally, one can define multilinear forms on ...
*
Homogeneous polynomial In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
*
Homogeneous function In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the '' ...
*
Tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tens ...
s


References

Multilinear algebra